Package 'fitODBOD'

Description

Lemmens , Knibbe and Tan(1988) described a study of self reported alcohol frequencies. The no of alcohol consumption data in two reference weeks is separately self reported by a randomly selected sample of 399 respondents in the Netherlands in 1983. Number of days a given individual consumes alcohol out of 7 days a week can be treated as a binomial variable. The collection of all such variables from all respondents would be defined as "Binomial Outcome Data".

Usage

Alcohol_data

Format

A data frame with 3 columns and 8 rows.

Days

No of Days Drunk

week1

Observed frequencies for week1

week2

Observed frequencies for week2

Source

Extracted from

Manoj, C., Wijekoon, P. & Yapa, R.D., 2013. The McDonald Generalized Beta-Binomial Distribution: A New Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling Overdispersion. International Journal of Statistics and Probability, 2(2), pp.24-41.

Available at: doi:10.5539/ijsp.v2n2p24

Examples

Alcohol_data$Days          # extracting the binomial random variables
sum(Alcohol_data$week2)       # summing all the frequencies in week2

Description

The below function has the ability to extract from the raw data to Binomial Outcome Data. This function simplifies the data into more presentable way to the user.

Usage

BODextract(data)

Arguments

Details

NOTE : If input parameters are not in given domain conditions necessary error messages will be provided to go further

Value

The output of BODextract gives a list format consisting

RV binomial random variables in vector form

Freq corresponding frequencies in vector form

Examples

datapoints <- sample(0:10,340,replace=TRUE) #creating a sample set of observations
BODextract(datapoints)                   #extracting binomial outcome data from observations
Random.variable <- BODextract(datapoints)$RV #extracting the binomial random variables

Description

Data in this example refer to 337 observations on the secondary association of chromosomes in Brassika; n , which is now the number of chromosomes, equals 3 and X is the number of pairs of bivalents showing association.

Usage

Chromosome_data

Format

A data frame with 2 columns and 4 rows

No.of.Asso

No of Associations

fre

Observed frequencies

Source

Extracted from

Paul, S.R., 1985. A three-parameter generalization of the binomial distribution. Communications in Statistics - Theory and Methods, 14(6), pp.1497-1506.

Available at: doi:10.1080/03610928508828990

Examples

Chromosome_data$No.of.Asso          #extracting the binomial random variables
sum(Chromosome_data$fre)            #summing all the frequencies

Description

The data refer to the numbers of courses taken by a class of 65 students from the first year of the Department of Statistics of Athens University of Economics. The students enrolled in this class attended 8 courses during the first year of their study. The total numbers of successful examinations (including resits) were recorded.

Usage

Course_data

Format

A data frame with 2 columns and 9 rows

sub.pass

subjects passed

fre

Observed frequencies

Source

Extracted from

Karlis, D. & Xekalaki, E., 2008. The Polygonal Distribution. In Advances in Mathematical and Statistical Modeling. Boston: Birkhuser Boston, pp. 21-33.

Available at: doi:10.1007/978-0-8176-4626-4_2.

Examples

Course_data$sub.pass             # extracting the binomial random variables
sum(Course_data$fre)             # summing all the frequencies

Description

These functions provide the ability for generating probability function values and cumulative probability function values for the Additive Binomial Distribution.

Usage

dAddBin(x,n,p,alpha)

Arguments

Details

The probability function and cumulative function can be constructed and are denoted below

The cumulative probability function is the summation of probability function values.

$P_{AddBin}(x)= {n \choose x} p^x (1-p)^{n-x}(\frac{alpha}{2}(\frac{x(x-1)}{p}+\frac{(n-x)(n-x-1)}{(1-p)}-\frac{alpha(n-1)n}{2})+1)$

The alpha is in between

$\frac{-2}{n(n-1)}min(\frac{p}{1-p},\frac{1-p}{p}) \le alpha \le (\frac{n+(2p-1)^2}{4p(1-p)})^{-1}$

$x = 0,1,2,3,...n$

$n = 1,2,3,...$

$0 < p < 1$

$-1 < alpha < 1$

The mean and the variance are denoted as

$E_{Addbin}[x]=np$

$Var_{Addbin}[x]=np(1-p)(1+(n-1)alpha)$

NOTE : If input parameters are not in given domain conditions necessary error messages will be provided to go further.

Value

The output of dAddBin gives a list format consisting

pdf probability function values in vector form.

mean mean of Additive Binomial Distribution.

var variance of Additive Binomial Distribution.

References

Johnson NL, Kemp AW, Kotz S (2005). Univariate discrete distributions, volume 444. John Wiley and Sons. Kupper LL, Haseman JK (1978). “The use of a correlated binomial model for the analysis of certain toxicological experiments.” Biometrics, 69–76. Paul SR (1985). “A three-parameter generalization of the binomial distribution.” History and Philosophy of Logic, 14(6), 1497–1506. Morel JG, Neerchal NK (2012). Overdispersion models in SAS. SAS Publishing.

Examples

#plotting the random variables and probability values
col <- rainbow(5)
a <- c(0.58,0.59,0.6,0.61,0.62)
b <- c(0.022,0.023,0.024,0.025,0.026)
plot(0,0,main="Additive binomial probability function graph",xlab="Binomial random variable",
ylab="Probability function values",xlim = c(0,10),ylim = c(0,0.5))
for (i in 1:5)
{
  lines(0:10,dAddBin(0:10,10,a[i],b[i])$pdf,col = col[i],lwd=2.85)
  points(0:10,dAddBin(0:10,10,a[i],b[i])$pdf,col = col[i],pch=16)
}

dAddBin(0:10,10,0.58,0.022)$pdf     #extracting the probability values
dAddBin(0:10,10,0.58,0.022)$mean    #extracting the mean
dAddBin(0:10,10,0.58,0.022)$var     #extracting the variance

#plotting the random variables and cumulative probability values
col <- rainbow(5)
a <- c(0.58,0.59,0.6,0.61,0.62)
b <- c(0.022,0.023,0.024,0.025,0.026)
plot(0,0,main="Additive binomial probability function graph",xlab="Binomial random variable",
ylab="Probability function values",xlim = c(0,10),ylim = c(0,1))
for (i in 1:5)
{
lines(0:10,pAddBin(0:10,10,a[i],b[i]),col = col[i],lwd=2.85)
points(0:10,pAddBin(0:10,10,a[i],b[i]),col = col[i],pch=16)
}

pAddBin(0:10,10,0.58,0.022)       #acquiring the cumulative probability values

Description

These functions provide the ability for generating probability density values, cumulative probability density values and moment about zero values for the Beta Distribution bounded between [0,1]

Usage

dBETA(p,a,b)

Arguments

Details

The probability density function and cumulative density function of a unit bounded Beta distribution with random variable P are given by

$g_{P}(p)= \frac{p^{a-1}(1-p)^{b-1}}{B(a,b)}$

; $0 \le p \le 1$

$G_{P}(p)= \frac{B_p(a,b)}{B(a,b)}$

; $0 \le p \le 1$

$a,b > 0$

The mean and the variance are denoted by

$E[P]= \frac{a}{a+b}$

$var[P]= \frac{ab}{(a+b)^2(a+b+1)}$

The moments about zero is denoted as

$E[P^r]= \prod_{i=0}^{r-1} (\frac{a+i}{a+b+i})$

$r = 1,2,3,...$

Defined as $B_p(a,b)=\int^p_0 t^{a-1} (1-t)^{b-1}\,dt$ is incomplete beta integrals and $B(a,b)$ is the beta function.

NOTE : If input parameters are not in given domain conditions necessary error messages will be provided to go further.

Value

The output of dBETA gives a list format consisting

pdf probability density values in vector form.

mean mean of the Beta distribution.

var variance of the Beta distribution.

References

Johnson NL, Kotz S, Balakrishnan N (1995). Continuous univariate distributions, volume 2, volume 289. John wiley and sons. Trenkler G (1996). “Continuous univariate distributions.” Computational Statistics and Data Analysis, 21(1), 119–119.

See Also

Beta

or

https://stat.ethz.ch/R-manual/R-devel/library/stats/html/Beta.html

Examples

#plotting the random variables and probability values
col <- rainbow(4)
a <- c(1,2,5,10)
plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
xlim = c(0,1),ylim = c(0,4))
for (i in 1:4)
{
lines(seq(0,1,by=0.01),dBETA(seq(0,1,by=0.01),a[i],a[i])$pdf,col = col[i])
}

dBETA(seq(0,1,by=0.01),2,3)$pdf   #extracting the pdf values
dBETA(seq(0,1,by=0.01),2,3)$mean  #extracting the mean
dBETA(seq(0,1,by=0.01),2,3)$var   #extracting the variance

#plotting the random variables and cumulative probability values
col <- rainbow(4)
a <- c(1,2,5,10)
plot(0,0,main="Cumulative density graph",xlab="Random variable",ylab="Cumulative density values",
xlim = c(0,1),ylim = c(0,1))
for (i in 1:4)
{
lines(seq(0,1,by=0.01),pBETA(seq(0,1,by=0.01),a[i],a[i]),col = col[i])
}

pBETA(seq(0,1,by=0.01),2,3)   #acquiring the cumulative probability values
mazBETA(1.4,3,2)              #acquiring the moment about zero values
mazBETA(2,3,2)-mazBETA(1,3,2)^2 #acquiring the variance for a=3,b=2

#only the integer value of moments is taken here because moments cannot be decimal
mazBETA(1.9,5.5,6)

Description

These functions provide the ability for generating probability function values and cumulative probability function values for the Beta-Binomial Distribution.

Usage

dBetaBin(x,n,a,b)

Arguments

Details

Mixing Beta distribution with Binomial distribution will create the Beta-Binomial distribution. The probability function and cumulative probability function can be constructed and are denoted below.

The cumulative probability function is the summation of probability function values.

$P_{BetaBin}(x)= {n \choose x} \frac{B(a+x,n+b-x)}{B(a,b)}$

$a,b > 0$

$x = 0,1,2,3,...n$

$n = 1,2,3,...$

The mean, variance and over dispersion are denoted as

$E_{BetaBin}[x]= \frac{na}{a+b}$

$Var_{BetaBin}[x]= \frac{(nab)}{(a+b)^2} \frac{(a+b+n)}{(a+b+1)}$

$over dispersion= \frac{1}{a+b+1}$

Defined as B(a,b) is the beta function.

Value

The output of dBetaBin gives a list format consisting

pdf probability function values in vector form.

mean mean of the Beta-Binomial Distribution.

var variance of the Beta-Binomial Distribution.

over.dis.para over dispersion value of the Beta-Binomial Distribution.

References

Young-Xu Y, Chan KA (2008). “Pooling overdispersed binomial data to estimate event rate.” BMC medical research methodology, 8, 1–12. Trenkler G (1996). “Continuous univariate distributions.” Computational Statistics and Data Analysis, 21(1), 119–119. HUGHES G, MADDEN L (1993). “Using the beta-binomial distribution to describe aggegated patterns of disease incidence.” Phytopathology, 83(7), 759–763.

Examples

#plotting the random variables and probability values
col <- rainbow(5)
a <- c(1,2,5,10,0.2)
plot(0,0,main="Beta-binomial probability function graph",xlab="Binomial random variable",
ylab="Probability function values",xlim = c(0,10),ylim = c(0,0.5))
for (i in 1:5)
{
lines(0:10,dBetaBin(0:10,10,a[i],a[i])$pdf,col = col[i],lwd=2.85)
points(0:10,dBetaBin(0:10,10,a[i],a[i])$pdf,col = col[i],pch=16)
}

dBetaBin(0:10,10,4,.2)$pdf    #extracting the pdf values
dBetaBin(0:10,10,4,.2)$mean   #extracting the mean
dBetaBin(0:10,10,4,.2)$var    #extracting the variance
dBetaBin(0:10,10,4,.2)$over.dis.para  #extracting the over dispersion value

#plotting the random variables and cumulative probability values
col <- rainbow(4)
a <- c(1,2,5,10)
plot(0,0,main="Cumulative probability function graph",xlab="Binomial random variable",
ylab="Cumulative probability function values",xlim = c(0,10),ylim = c(0,1))
for (i in 1:4)
{
lines(0:10,pBetaBin(0:10,10,a[i],a[i]),col = col[i])
points(0:10,pBetaBin(0:10,10,a[i],a[i]),col = col[i])
}

pBetaBin(0:10,10,4,.2)   #acquiring the cumulative probability values

Description

These functions provide the ability for generating probability function values and cumulative probability function values for the Beta-Correlated Binomial Distribution.

Usage

dBetaCorrBin(x,n,cov,a,b)

Arguments

Details

The probability function and cumulative function can be constructed and are denoted below

The cumulative probability function is the summation of probability function values.

$x = 0,1,2,3,...n$

$n = 1,2,3,...$

$0 < a,b$

$-\infty < cov < +\infty$

$0 < p < 1$

$p=\frac{a}{a+b}$

$\Theta=\frac{1}{a+b}$

The Correlation is in between

$\frac{-2}{n(n-1)} min(\frac{p}{1-p},\frac{1-p}{p}) \le correlation \le \frac{2p(1-p)}{(n-1)p(1-p)+0.25-fo}$

where $fo=min [(x-(n-1)p-0.5)^2]$

The mean and the variance are denoted as

$E_{BetaCorrBin}[x]= np$

$Var_{BetaCorrBin}[x]= np(1-p)(n\Theta+1)(1+\Theta)^{-1}+n(n-1)cov$

$Corr_{BetaCorrBin}[x]=\frac{cov}{p(1-p)}$

NOTE : If input parameters are not in given domain conditions necessary error messages will be provided to go further.

Value

The output of dBetaCorrBin gives a list format consisting

pdf probability function values in vector form.

mean mean of Beta-Correlated Binomial Distribution.

var variance of Beta-Correlated Binomial Distribution.

corr correlation of Beta-Correlated Binomial Distribution.

mincorr minimum correlation value possible.

maxcorr maximum correlation value possible.

References

Paul SR (1985). “A three-parameter generalization of the binomial distribution.” History and Philosophy of Logic, 14(6), 1497–1506.

Examples

#plotting the random variables and probability values
col <- rainbow(5)
a <- c(9.0,10,11,12,13)
b <- c(8.0,8.1,8.2,8.3,8.4)
plot(0,0,main="Beta-Correlated binomial probability function graph",xlab="Binomial random variable",
ylab="Probability function values",xlim = c(0,10),ylim = c(0,0.5))
for (i in 1:5)
{
lines(0:10,dBetaCorrBin(0:10,10,0.001,a[i],b[i])$pdf,col = col[i],lwd=2.85)
points(0:10,dBetaCorrBin(0:10,10,0.001,a[i],b[i])$pdf,col = col[i],pch=16)
}

dBetaCorrBin(0:10,10,0.001,10,13)$pdf      #extracting the pdf values
dBetaCorrBin(0:10,10,0.001,10,13)$mean     #extracting the mean
dBetaCorrBin(0:10,10,0.001,10,13)$var      #extracting the variance
dBetaCorrBin(0:10,10,0.001,10,13)$corr     #extracting the correlation
dBetaCorrBin(0:10,10,0.001,10,13)$mincorr  #extracting the minimum correlation value
dBetaCorrBin(0:10,10,0.001,10,13)$maxcorr  #extracting the maximum correlation value

#plotting the random variables and cumulative probability values
col <- rainbow(5)
a <- c(9.0,10,11,12,13)
b <- c(8.0,8.1,8.2,8.3,8.4)
plot(0,0,main="Beta-Correlated binomial probability function graph",xlab="Binomial random variable",
ylab="Probability function values",xlim = c(0,10),ylim = c(0,1))
for (i in 1:5)
{
lines(0:10,pBetaCorrBin(0:10,10,0.001,a[i],b[i]),col = col[i],lwd=2.85)
points(0:10,pBetaCorrBin(0:10,10,0.001,a[i],b[i]),col = col[i],pch=16)
}

pBetaCorrBin(0:10,10,0.001,10,13)      #acquiring the cumulative probability values

Description

These functions provide the ability for generating probability function values and cumulative probability function values for the COM Poisson Binomial Distribution.

Usage

dCOMPBin(x,n,p,v)

Arguments

Details

The probability function and cumulative function can be constructed and are denoted below

The cumulative probability function is the summation of probability function values.

$P_{COMPBin}(x) = \frac{{n \choose x}^v p^x (1-p)^{n-x}}{\sum_{j=0}^{n} {n \choose j}^v p^j (1-p)^{(n-j)}}$

$x = 0,1,2,3,...n$

$n = 1,2,3,...$

$0 < p < 1$

$-\infty < v < +\infty$

NOTE : If input parameters are not in given domain conditions necessary error messages will be provided to go further.

Value

The output of dCOMPBin gives a list format consisting

pdf probability function values in vector form.

mean mean of COM Poisson Binomial Distribution.

var variance of COM Poisson Binomial Distribution.

References

Borges P, Rodrigues J, Balakrishnan N, Bazan J (2014). “A COM–Poisson type generalization of the binomial distribution and its properties and applications.” Statistics and Probability Letters, 87, 158–166.

Examples

#plotting the random variables and probability values
col <- rainbow(5)
a <- c(0.58,0.59,0.6,0.61,0.62)
b <- c(0.022,0.023,0.024,0.025,0.026)
plot(0,0,main="COM Poisson Binomial probability function graph",xlab="Binomial random variable",
ylab="Probability function values",xlim = c(0,10),ylim = c(0,0.5))
for (i in 1:5)
{
lines(0:10,dCOMPBin(0:10,10,a[i],b[i])$pdf,col = col[i],lwd=2.85)
points(0:10,dCOMPBin(0:10,10,a[i],b[i])$pdf,col = col[i],pch=16)
}

dCOMPBin(0:10,10,0.58,0.022)$pdf      #extracting the pdf values
dCOMPBin(0:10,10,0.58,0.022)$mean     #extracting the mean
dCOMPBin(0:10,10,0.58,0.022)$var      #extracting the variance

#plotting the random variables and cumulative probability values
col <- rainbow(5)
a <- c(0.58,0.59,0.6,0.61,0.62)
b <- c(0.022,0.023,0.024,0.025,0.026)
plot(0,0,main="COM Poisson Binomial probability function graph",xlab="Binomial random variable",
ylab="Probability function values",xlim = c(0,10),ylim = c(0,1))
for (i in 1:5)
{
lines(0:10,pCOMPBin(0:10,10,a[i],b[i]),col = col[i],lwd=2.85)
points(0:10,pCOMPBin(0:10,10,a[i],b[i]),col = col[i],pch=16)
}

pCOMPBin(0:10,10,0.58,0.022)      #acquiring the cumulative probability values

Description

These functions provide the ability for generating probability function values and cumulative probability function values for the Correlated Binomial Distribution.

Usage

dCorrBin(x,n,p,cov)

Arguments

Details

The probability function and cumulative function can be constructed and are denoted below

The cumulative probability function is the summation of probability function values.

$P_{CorrBin}(x) = {n \choose x}(p^x)(1-p)^{n-x}(1+(\frac{cov}{2p^2(1-p)^2})((x-np)^2+x(2p-1)-np^2))$

$x = 0,1,2,3,...n$

$n = 1,2,3,...$

$0 < p < 1$

$-\infty < cov < +\infty$

The Correlation is in between

$\frac{-2}{n(n-1)} min(\frac{p}{1-p},\frac{1-p}{p}) \le correlation \le \frac{2p(1-p)}{(n-1)p(1-p)+0.25-fo}$

where $fo=min [(x-(n-1)p-0.5)^2]$

The mean and the variance are denoted as

$E_{CorrBin}[x]= np$

$Var_{CorrBin}[x]= n(p(1-p)+(n-1)cov)$

$Corr_{CorrBin}[x]=\frac{cov}{p(1-p)}$

NOTE : If input parameters are not in given domain conditions necessary error messages will be provided to go further.

Value

The output of dCorrBin gives a list format consisting

pdf probability function values in vector form.

mean mean of Correlated Binomial Distribution.

var variance of Correlated Binomial Distribution.

corr correlation of Correlated Binomial Distribution.

mincorr minimum correlation value possible.

maxcorr maximum correlation value possible.

References

Johnson NL, Kemp AW, Kotz S (2005). Univariate discrete distributions, volume 444. John Wiley and Sons. Kupper LL, Haseman JK (1978). “The use of a correlated binomial model for the analysis of certain toxicological experiments.” Biometrics, 69–76. Paul SR (1985). “A three-parameter generalization of the binomial distribution.” History and Philosophy of Logic, 14(6), 1497–1506. Morel JG, Neerchal NK (2012). Overdispersion models in SAS. SAS Publishing.

Examples

#plotting the random variables and probability values
col <- rainbow(5)
a <- c(0.58,0.59,0.6,0.61,0.62)
b <- c(0.022,0.023,0.024,0.025,0.026)
plot(0,0,main="Correlated binomial probability function graph",xlab="Binomial random variable",
ylab="Probability function values",xlim = c(0,10),ylim = c(0,0.5))
for (i in 1:5)
{
lines(0:10,dCorrBin(0:10,10,a[i],b[i])$pdf,col = col[i],lwd=2.85)
points(0:10,dCorrBin(0:10,10,a[i],b[i])$pdf,col = col[i],pch=16)
}

dCorrBin(0:10,10,0.58,0.022)$pdf      #extracting the pdf values
dCorrBin(0:10,10,0.58,0.022)$mean     #extracting the mean
dCorrBin(0:10,10,0.58,0.022)$var      #extracting the variance
dCorrBin(0:10,10,0.58,0.022)$corr     #extracting the correlation
dCorrBin(0:10,10,0.58,0.022)$mincorr  #extracting the minimum correlation value
dCorrBin(0:10,10,0.58,0.022)$maxcorr  #extracting the maximum correlation value

#plotting the random variables and cumulative probability values
col <- rainbow(5)
a <- c(0.58,0.59,0.6,0.61,0.62)
b <- c(0.022,0.023,0.024,0.025,0.026)
plot(0,0,main="Correlated binomial probability function graph",xlab="Binomial random variable",
ylab="Probability function values",xlim = c(0,10),ylim = c(0,1))
for (i in 1:5)
{
lines(0:10,pCorrBin(0:10,10,a[i],b[i]),col = col[i],lwd=2.85)
points(0:10,pCorrBin(0:10,10,a[i],b[i]),col = col[i],pch=16)
}

pCorrBin(0:10,10,0.58,0.022)      #acquiring the cumulative probability values

Description

These functions provide the ability for generating probability density values, cumulative probability density values and moment about zero values for Gamma Distribution bounded between [0,1].

Usage

dGAMMA(p,c,l)

Arguments

Details

The probability density function and cumulative density function of a unit bounded Gamma distribution with random variable P are given by

$g_{P}(p) = \frac{ c^l p^{c-1}}{\gamma(l)} [ln(1/p)]^{l-1}$

; $0 \le p \le 1$

$G_{P}(p) = \frac{ Ig(l,cln(1/p))}{\gamma(l)}$

; $0 \le p \le 1$

$l,c > 0$

The mean the variance are denoted by

$E[P] = (\frac{c}{c+1})^l$

$var[P] = (\frac{c}{c+2})^l - (\frac{c}{c+1})^{2l}$

The moments about zero is denoted as

$E[P^r]=(\frac{c}{c+r})^l$

$r = 1,2,3,...$

Defined as $\gamma(l)$ is the gamma function Defined as $Ig(l,cln(1/p))= \int_0^{cln(1/p)} t^{l-1} e^{-t}dt$ is the Lower incomplete gamma function

NOTE : If input parameters are not in given domain conditions necessary error messages will be provided to go further.

Value

The output of dGAMMA gives a list format consisting

pdf probability density values in vector form.

mean mean of the Gamma distribution.

var variance of Gamma distribution.

References

Olshen AC (1938). “Transformations of the pearson type III distribution.” The Annals of Mathematical Statistics, 9(3), 176–200.

See Also

GammaDist

Examples

#plotting the random variables and probability values
col <- rainbow(4)
a <- c(1,2,5,10)
plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
xlim = c(0,1),ylim = c(0,4))
for (i in 1:4)
{
lines(seq(0,1,by=0.01),dGAMMA(seq(0,1,by=0.01),a[i],a[i])$pdf,col = col[i])
}

dGAMMA(seq(0,1,by=0.01),5,6)$pdf   #extracting the pdf values
dGAMMA(seq(0,1,by=0.01),5,6)$mean  #extracting the mean
dGAMMA(seq(0,1,by=0.01),5,6)$var   #extracting the variance

#plotting the random variables and cumulative probability values
col <- rainbow(4)
a <- c(1,2,5,10)
plot(0,0,main="Cumulative density graph",xlab="Random variable",ylab="Cumulative density values",
xlim = c(0,1),ylim = c(0,1))
for (i in 1:4)
{
lines(seq(0,1,by=0.01),pGAMMA(seq(0,1,by=0.01),a[i],a[i]),col = col[i])
}

pGAMMA(seq(0,1,by=0.01),5,6)   #acquiring the cumulative probability values
mazGAMMA(1.4,5,6)              #acquiring the moment about zero values
mazGAMMA(2,5,6)-mazGAMMA(1,5,6)^2 #acquiring the variance for a=5,b=6

#only the integer value of moments is taken here because moments cannot be decimal
mazGAMMA(1.9,5.5,6)

Description

These functions provide the ability for generating probability function values and cumulative probability function values for the Gamma Binomial Distribution.

Usage

dGammaBin(x,n,c,l)

Arguments

Details

Mixing Gamma distribution with Binomial distribution will create the the Gamma Binomial distribution. The probability function and cumulative probability function can be constructed and are denoted below.

The cumulative probability function is the summation of probability function values.

$P_{GammaBin}[x]= {n \choose x} \sum_{j=0}^{n-x} {n-x \choose j} (-1)^j (\frac{c}{c+x+j})^l$

$c,l > 0$

$x = 0,1,2,...,n$

$n = 1,2,3,...$

The mean, variance and over dispersion are denoted as

$E_{GammaBin}[x] = (\frac{c}{c+1})^l$

$Var_{GammaBin}[x] = n^2[(\frac{c}{c+2})^l - (\frac{c}{c+1})^{2l}] + n(\frac{c}{c+1})^l{1-)(\frac{c+1}{c+2})^l}$

$over dispersion= \frac{(\frac{c}{c+2})^l - (\frac{c}{c+1})^{2l}}{(\frac{c}{c+1})^l[1-(\frac{c}{c+1})^l]}$

Value

The output of dGammaBin gives a list format consisting

pdf probability function values in vector form.

mean mean of the Gamma Binomial Distribution.

var variance of the Gamma Binomial Distribution.

over.dis.para over dispersion value of the Gamma Binomial Distribution.

References

Grassia A (1977). “On a family of distributions with argument between 0 and 1 obtained by transformation of the gamma and derived compound distributions.” Australian Journal of Statistics, 19(2), 108–114.

Examples

#plotting the random variables and probability values
col <- rainbow(5)
a <- c(1,2,5,10,0.2)
plot(0,0,main="Gamma Binomial probability function graph",xlab="Binomial random variable",
ylab="Probability function values",xlim = c(0,10),ylim = c(0,0.5))
for (i in 1:5)
{
lines(0:10,dGammaBin(0:10,10,a[i],a[i])$pdf,col = col[i],lwd=2.85)
points(0:10,dGammaBin(0:10,10,a[i],a[i])$pdf,col = col[i],pch=16)
}

dGammaBin(0:10,10,4,.2)$pdf    #extracting the pdf values
dGammaBin(0:10,10,4,.2)$mean   #extracting the mean
dGammaBin(0:10,10,4,.2)$var    #extracting the variance
dGammaBin(0:10,10,4,.2)$over.dis.para  #extracting the over dispersion value

#plotting the random variables and cumulative probability values
col <- rainbow(4)
a <- c(1,2,5,10)
plot(0,0,main="Cumulative probability function graph",xlab="Binomial random variable",
ylab="Cumulative probability function values",xlim = c(0,10),ylim = c(0,1))
for (i in 1:4)
{
lines(0:10,pGammaBin(0:10,10,a[i],a[i]),col = col[i])
points(0:10,pGammaBin(0:10,10,a[i],a[i]),col = col[i])
}

pGammaBin(0:10,10,4,.2)   #acquiring the cumulative probability values

Description

These functions provide the ability for generating probability density values, cumulative probability density values and moment about zero values for the Generalized Beta Type-1 Distribution bounded between [0,1].

Usage

dGBeta1(p,a,b,c)

Arguments

Details

The probability density function and cumulative density function of a unit bounded Generalized Beta Type-1 Distribution with random variable P are given by

$g_{P}(p)= \frac{c}{B(a,b)} p^{ac-1} (1-p^c)^{b-1}$

; $0 \le p \le 1$

$G_{P}(p)= \frac{p^{ac}}{aB(a,b)} 2F1(a,1-b;p^c;a+1)$

$0 \le p \le 1$

$a,b,c > 0$

The mean and the variance are denoted by

$E[P]= \frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})}$

$var[P]= \frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2$

The moments about zero is denoted as

$E[P^r]= \frac{B(a+b,\frac{r}{c})}{B(a,\frac{r}{c})}$

$r = 1,2,3,....$

Defined as $B(a,b)$ is Beta function. Defined as $2F1(a,b;c;d)$ is Gaussian Hypergeometric function.

NOTE : If input parameters are not in given domain conditions necessary error messages will be provided to go further.

Value

The output of dGBeta1 gives a list format consisting

pdf probability density values in vector form.

mean mean of the Generalized Beta Type-1 Distribution.

var variance of the Generalized Beta Type-1 Distribution.

References

Manoj C, Wijekoon P, Yapa RD (2013). “The McDonald generalized beta-binomial distribution: A new binomial mixture distribution and simulation based comparison with its nested distributions in handling overdispersion.” International journal of statistics and probability, 2(2), 24. Janiffer NM, Islam A, Luke O, others (2014). “Estimating Equations for Estimation of Mcdonald Generalized Beta—Binomial Parameters.” Open Journal of Statistics, 4(09), 702. Roozegar R, Tahmasebi S, Jafari AA (2017). “The McDonald Gompertz distribution: properties and applications.” Communications in Statistics-Simulation and Computation, 46(5), 3341–3355.

Examples

#plotting the random variables and probability values
col <- rainbow(5)
a <- c(.1,.2,.3,1.5,2.15)
plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
xlim = c(0,1),ylim = c(0,10))
for (i in 1:5)
{
lines(seq(0,1,by=0.001),dGBeta1(seq(0,1,by=0.001),a[i],1,2*a[i])$pdf,col = col[i])
}

dGBeta1(seq(0,1,by=0.01),2,3,1)$pdf    #extracting the pdf values
dGBeta1(seq(0,1,by=0.01),2,3,1)$mean   #extracting the mean
dGBeta1(seq(0,1,by=0.01),2,3,1)$var    #extracting the variance

pGBeta1(0.04,2,3,4)        #acquiring the cdf values for a=2,b=3,c=4
mazGBeta1(1.4,3,2,2)              #acquiring the moment about zero values
mazGBeta1(2,3,2,2)-mazGBeta1(1,3,2,2)^2        #acquiring the variance for a=3,b=2,c=2

#only the integer value of moments is taken here because moments cannot be decimal
mazGBeta1(3.2,3,2,2)

Description

These functions provide the ability for generating probability function values and cumulative probability function values for the Gaussian Hypergeometric Generalized Beta Binomial distribution.

Usage

dGHGBB(x,n,a,b,c)

Arguments

Details

Mixing Gaussian Hypergeometric Generalized Beta distribution with Binomial distribution will create the Gaussian Hypergeometric Generalized Beta Binomial distribution. The probability function and cumulative probability function can be constructed and are denoted below.

The cumulative probability function is the summation of probability function values.

$P_{GHGBB}(x)=\frac{1}{2F1(-n,a;-b-n+1;c)} {n \choose x} \frac{B(x+a,n-x+b)}{B(a,b+n)}(c^x)$

$a,b,c > 0$

$x = 0,1,2,...n$

$n = 1,2,3,...$

The mean, variance and over dispersion are denoted as

$E_{GHGBB}[x]= nE_{GHGBeta}$

$Var_{GHGBB}[x]= nE_{GHGBeta}(1-E_{GHGBeta})+ n(n-1)Var_{GHGBeta}$

$over dispersion= \frac{var_{GHGBeta}}{E_{GHGBeta}(1-E_{GHGBeta})}$

Defined as $B(a,b)$ is the beta function. Defined as $2F1(a,b;c;d)$ is the Gaussian Hypergeometric function

NOTE : If input parameters are not in given domain conditions necessary error messages will be provided to go further.

Value

The output of dGHGBB gives a list format consisting

pdf probability function values in vector form.

mean mean of Gaussian Hypergeometric Generalized Beta Binomial Distribution.

var variance of Gaussian Hypergeometric Generalized Beta Binomial Distribution.

over.dis.para over dispersion value of Gaussian Hypergeometric Generalized Beta Binomial Distribution.

References

Rodriguez-Avi J, Conde-Sanchez A, Saez-Castillo AJ, Olmo-Jimenez MJ (2007). “A generalization of the beta–binomial distribution.” Journal of the Royal Statistical Society Series C: Applied Statistics, 56(1), 51–61. Pearson JW (2009). Computation of hypergeometric functions. Ph.D. thesis, University of Oxford.

See Also

hypergeo_powerseries

Examples

#plotting the random variables and probability values
col <- rainbow(6)
a <- c(.1,.2,.3,1.5,2.1,3)
plot(0,0,main="GHGBB probability function graph",xlab="Binomial random variable",
ylab="Probability function values",xlim = c(0,7),ylim = c(0,0.9))
for (i in 1:6)
{
lines(0:7,dGHGBB(0:7,7,1+a[i],0.3,1+a[i])$pdf,col = col[i],lwd=2.85)
points(0:7,dGHGBB(0:7,7,1+a[i],0.3,1+a[i])$pdf,col = col[i],pch=16)
}

dGHGBB(0:7,7,1.3,0.3,1.3)$pdf      #extracting the pdf values
dGHGBB(0:7,7,1.3,0.3,1.3)$mean     #extracting the mean
dGHGBB(0:7,7,1.3,0.3,1.3)$var      #extracting the variance
dGHGBB(0:7,7,1.3,0.3,1.3)$over.dis.par  #extracting the over dispersion value

#plotting the random variables and cumulative probability values
col <- rainbow(4)
a <- c(1,2,5,10)
plot(0,0,main="Cumulative probability function graph",xlab="Binomial random variable",
ylab="Cumulative probability function values",xlim = c(0,7),ylim = c(0,1))
for (i in 1:4)
{
lines(0:7,pGHGBB(0:7,7,1+a[i],0.3,1+a[i]),col = col[i])
points(0:7,pGHGBB(0:7,7,1+a[i],0.3,1+a[i]),col = col[i])
}

pGHGBB(0:7,7,1.3,0.3,1.3)     #acquiring the cumulative probability values

Description

These functions provide the ability for generating probability density values, cumulative probability density values and moment about zero values for the Gaussian Hypergeometric Generalized Beta distribution bounded between [0,1].

Usage

dGHGBeta(p,n,a,b,c)

Arguments

Details

The probability density function and cumulative density function of a unit bounded Gaussian Hypergeometric Generalized Beta Distribution with random variable P are given by

$g_{P}(p)= \frac{1}{B(a,b)}\frac{2F1(-n,a;-b-n+1;1)}{2F1(-n,a;-b-n+1;c)} p^{a-1}(1-p)^{b-1} \frac{c^{b+n}}{(c+(1-c)p)^{a+b+n}}$

; $0 \le p \le 1$

$G_{P}(p)= \int^p_0 \frac{1}{B(a,b)}\frac{2F1(-n,a;-b-n+1;1)}{2F1(-n,a;-b-n+1;c)} t^{a-1}(1-t)^{b-1}\frac{c^{b+n}}{(c+(1-c)t)^{a+b+n}} \,dt$

; $0 \le p \le 1$

$a,b,c > 0$

$n = 1,2,3,...$

The mean and the variance are denoted by

$E[P]= \int^1_0 \frac{p}{B(a,b)}\frac{2F1(-n,a;-b-n+1;1)}{2F1(-n,a;-b-n+1;c)} p^{a-1}(1-p)^{b-1}\frac{c^{b+n}}{(c+(1-c)p)^{a+b+n}} \,dp$

$var[P]= \int^1_0 \frac{p^2}{B(a,b)}\frac{2F1(-n,a;-b-n+1;1)}{2F1(-n,a;-b-n+1;c)} p^{a-1}(1-p)^{b-1}\frac{c^{b+n}}{(c+(1-c)p)^{a+b+n}} \,dp - (E[p])^2$

The moments about zero is denoted as

$E[P^r]= \int^1_0 \frac{p^r}{B(a,b)}\frac{2F1(-n,a;-b-n+1;1)}{2F1(-n,a;-b-n+1;c)} p^{a-1}(1-p)^{b-1}\frac{c^{b+n}}{(c+(1-c)p)^{a+b+n}} \,dp$

$r = 1,2,3,...$

Defined as $B(a,b)$ as the beta function. Defined as $2F1(a,b;c;d)$ as the Gaussian Hypergeometric function.

NOTE : If input parameters are not in given domain conditions necessary error messages will be provided to go further.

Value

The output of dGHGBeta gives a list format consisting

pdf probability density values in vector form.

mean mean of the Gaussian Hypergeometric Generalized Beta Distribution.

var variance of the Gaussian Hypergeometric Generalized Beta Distribution.

References

Rodriguez-Avi J, Conde-Sanchez A, Saez-Castillo AJ, Olmo-Jimenez MJ (2007). “A generalization of the beta–binomial distribution.” Journal of the Royal Statistical Society Series C: Applied Statistics, 56(1), 51–61. Pearson JW (2009). Computation of hypergeometric functions. Ph.D. thesis, University of Oxford.

See Also

hypergeo_powerseries

Examples

#plotting the random variables and probability values
col <- rainbow(5)
a <- c(.1,.2,.3,1.5,2.15)
plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
xlim = c(0,1),ylim = c(0,10))
for (i in 1:5)
{
lines(seq(0,1,by=0.001),dGHGBeta(seq(0,1,by=0.001),7,1+a[i],0.3,1+a[i])$pdf,col = col[i])
}

dGHGBeta(seq(0,1,by=0.01),7,1.6312,0.3913,0.6659)$pdf   #extracting the pdf values
dGHGBeta(seq(0,1,by=0.01),7,1.6312,0.3913,0.6659)$mean  #extracting the mean
dGHGBeta(seq(0,1,by=0.01),7,1.6312,0.3913,0.6659)$var   #extracting the variance

#plotting the random variables and cumulative probability values
col <- rainbow(6)
a <- c(.1,.2,.3,1.5,2.1,3)
plot(0,0,main="Cumulative density graph",xlab="Random variable",ylab="Cumulative density values",
xlim = c(0,1),ylim = c(0,1))
for (i in 1:6)
{
lines(seq(0.01,1,by=0.001),pGHGBeta(seq(0.01,1,by=0.001),7,1+a[i],0.3,1+a[i]),col=col[i])
}

pGHGBeta(seq(0,1,by=0.01),7,1.6312,0.3913,0.6659) #acquiring the cumulative probability values
mazGHGBeta(1.4,7,1.6312,0.3913,0.6659)            #acquiring the moment about zero values

#acquiring the variance for a=1.6312,b=0.3913,c=0.6659
mazGHGBeta(2,7,1.6312,0.3913,0.6659)-mazGHGBeta(1,7,1.6312,0.3913,0.6659)^2

#only the integer value of moments is taken here because moments cannot be decimal
mazGHGBeta(1.9,15,5,6,1)

Description

These functions provide the ability for generating probability function values and cumulative probability function values for the Grassia-II-Binomial Distribution.

Usage

dGrassiaIIBin(x,n,a,b)

Arguments

Details

Mixing Gamma distribution with Binomial distribution will create the the Grassia-II-Binomial distribution, only when (1-p)=e^(-lambda) of the Binomial distribution. The probability function and cumulative probability function can be constructed and are denoted below.

The cumulative probability function is the summation of probability function values.

$P_{GrassiaIIBin}[x]= {n \choose x} \sum_{j=0}^{x} {x \choose j} (-1)^{x-j} (1+b(n-j))^{-a}$

$a,b > 0$

$x = 0,1,2,...,n$